Perfect Competition in a Bilateral Monopoly (In Honor of Martin S

Keywords: Limit orders, double auction, Nash equilibria, Walrasequilibria, perfect competition, bilateral monopoly, mechanism designJEL Classification: C72, D41, D42, D44, D61 ∗It is a p

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PERFECT COMPETITION IN A BILATERAL MONOPOLY

(In Honor of Martin Shubik)

By Pradeep Dubey and Dieter Sondermann

September 2005

COWLES FOUNDATION DISCUSSION PAPER NO 1534

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS

YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/

PERFECT COMPETITION IN A

BILATERAL MONOPOLY

(In honor of Martin Shubik ∗ )

March 21, 2005

Abstract

We show that if limit orders are required to vary smoothly, thenstrategic (Nash) equilibria of the double auction mechanism yield com-petitive (Walras) allocations It is not necessary to have competitors

on any side of any market: smooth trading is a substitute for pricewars In particular, Nash equilibria are Walrasian even in a bilateralmonopoly

Keywords: Limit orders, double auction, Nash equilibria, Walrasequilibria, perfect competition, bilateral monopoly, mechanism designJEL Classification: C72, D41, D42, D44, D61

∗It is a pleasure for us to dedicate this paper to Martin Shubik who founded and developed (in collaboration with others, particularly Lloyd Shapley) the field of Strategic Market Games in a general equilibrium framework

†Center for Game Theory, Dept of Economics, SUNY at Stony Brook and Cowles Foundation, Yale University

‡Department of Economics, University of Bonn, Bonn.

1 Introduction

As is well-known Walrasian analysis is built upon the Hypothesis of PerfectCompetition, which can be taken as in Mas-Colell (1980) to state: “ thatprices are publicly quoted and are viewed by the economic agents as ex-ogenously given” Attempts to go beyond Walrasian analysis have in par-ticular involved giving “a theoretical explanation of the Hypothesis itself”(Mas-Colell (1980)) Among these the most remarkable are without doubtthe 19th century contributions of Bertrand, Cournot and Edgeworth (for anoverview, see Stigler (1965)) The Cournot approach was explored inten-sively, in a general equilibrium framework, in the symposium issue entitled

“Non-cooperative Approaches to the Theory of Perfect Competition” nal of Economic Theory, Vol 22 (1980))

(Jour-The features common to most of the symposium articles are:

(a) The strategies employed by the agents are of the Cournot type, i.e.,consist in quoting quantities

(b) The (insignificant) size of any agent relative to the market is the keyexplanatory variable for the tendency of strategic behavior to approx-imate perfect competition and, in its wake, to lead to Walrasian out-comes (Mas-Colell (1980), p.122)

The extension of pure quantity strategies from Cournot’s partial rium model of oligopoly to a general equilibrium framework, however, doesraise questions Underlying the Cournot model is a demand curve for theparticular market under consideration which enables the suppliers to relatequantities, via prices, to expected receipts If such a close relationship is notprovided by the market, then it seems more natural to us that an agent will

equilib-no longer confine himself to quoting quantities, i.e., to pure buy-or-sell ket orders To protect himself against “market uncertainty - or illiquidity, ormanipulation by other agents1”, he will also quote prices limiting the execu-

mar-tion of those orders, consenting to sell q units of commodity j only if its price

is p or more, or buy ˜ q units only if its price is ˜ p or less By sending multiple

1 to quote from Mertens (2003)

orders of this kind an agent can approximate any monotone demand or ply curve in a market by a step function, as was done in Dubey (1982, 1994).Here we go further and give each agent full manoeuvrability He places acontinuum of infinitesimal limit-price orders, which in effect enables him tosend any monotone, continuous demand or supply curve for each commodity.The upshot is a striking result: provided only that all commodity marketsare “active” (i.e there is positive trade in them), and no matter how thin

sup-they are, strategic (Nash) equilibria (SE) coincide - in outcome space - with

competitive (Walras) equilibria (CE) Our result thus provides a rationale,

based on strategic competition, for Walrasian outcomes even in the case of abilateral monopoly This brings it in sharp contrast to Dubey (1982, 1994),where it was necessary to have competition on both sides of each market (inthe sense of there being at least two active buyers and two active sellers foreach commodity) in order to conclude that SE are CE

The models in Dubey (1982, 1994) rely on competition that is throat” in the spirit of Betrand Any agent can take over a whole chunk

“cut-of some buy (sell) order from another by quoting an infinitesimally higher(lower) price Our model is not based on the possibility of such takeovers.Instead it requires that agents’ behavior be “smooth”, with commoditiesbought (sold) in infinitesimal increments of continuously non-increasing (non-decreasing) prices The key point of our paper is that such smooth trading

is a substitute for cut-throat price wars, and also gives rise to Walrasianoutcomes A monopolist may be in sole command of his own resource, butnevertheless he will be reduced to behaving as if he had cut-throat rivals,once smooth trading sets in A related phenomenon2 was analyzed in Coase(1972) (and following Coase (1972), a long line of literature, see e.g., Bulow(1982), Gaskins (1974), Schmalensee (1979)) There, too, a monopolist wasshown to forfeit his power, but this happened in the setting of durable goodswhich could be sold sequentially over time to infinitely patient customers Inour model the monopolist loses power even with perishable goods which aretraded at one instant of time But we do need, unlike Coase, smooth strategicbehavior on both sides of every market as well as convex preferences

It must be emphasized that our model is based on decentralized markets.

2 We thank John Geanakoplos for this reference.

Each commodity j is traded against fiat money (“unit of account”), and orders sent to the markets k 6= j for other commodities k, do not affect how market j functions Thus we do not allow an agent to link his buy-

order for a commodity to whether the sell-order for another commodity goesthrough.3 The only connection between different commodity markets is thebudget-constraint of agents, requiring them to cover purchases out of theirsales receipts Our model is therefore an order-of-magnitude simpler thanthat of Mertens (2003), where cross-market limit orders are permitted Inspite of this paucity of our strategy-space compared to Mertens (2003), weexactly implement 4 CE via our mechanism (modulo activity in markets) Incontrast, SE form a large superset 5 of CE in Mertens (2003) (though, wehasten to add, the implementation of CE was never the aim there, rather itwas to well-define a mechanism that allowed for a rich menu of cross-marketlimit-orders)

For better perspective, we consider two somewhat contrasting versions ofour model In the first version agents act under the optimistic illusion thatthey can exert perfect price discrimination: sell to others, starting at thehighest quoted market price (or buy, starting at the lowest) The equilibriumpoint (EP) that we define does not correspond to a strategic equilibrium (SE)

of a standard game, because we allow agents to speculate that they couldtrade at much better prices, via unilateral deviations, than any proper gameform would permit Nevertheless we think that EP is an interesting concept

in its own right

In the second version we turn to a standard market game, akin to that

of Dubey (1982) and Dubey (1994) Here each agent is grimly realistic andrealizes that he will be able to buy (sell) only after higher-priced buyers(lower-priced sellers) have been serviced at the market, and that the prices

he gets are apropos his own quotations, not the best going.6 To

accommo-3 That would be like allowing agents to submit demand functions based on the whole price vector.

4 Indeed, our result may be interpreted in terms of the mechanism design literature (see Section 3).

5 For instance, the SE of Shapley’s “windows model” (see Sahi and Yao (1989)) are also

SE in Mertens’ model.

6 We could make the same assumption also in the first market model However we would

date economies in which CE consumptions could occur on the boundary7, itbecomes needful here to introduce a “market maker” who has infinitesimalinventories of every good, and stands ready to provide them if sellers renege

on their promises of delivery It turns out that, at our SE, the market maker

is never active But it is important for agents to imagine his presence whenthey think about what they could get were they to unilaterally deviate.Though the two versions are built on quite different behaviorial hypothe-ses, we find their equilibria (the EP and the SE) lead to the same outcomes,namely Walrasian

Our model shares some of the weaknesses of the Walrasian models Inparticular, since it is based on the static concept of a strategic equilibrium,our model does not address the question of what dynamic forces bring theequilibrium about and ensure that individual strategic plans become jointlycompatible But it goes beyond the Walrasian notion in at least three im-portant ways:

(a) It is not assumed that the economic agents face perfectly elastic supplyand demand curves

(b) Prices are not quoted from outside but set by the agents themselves.Each agent, operating in a market, realizes and exerts his ability toinfluence price

(c) Strategies of the individuals (i.e supply and demand curves submitted

to the market) need not be based on their true characteristics ences and endowments)

(prefer-lose economic insight, as to what happens to the consumers’ and producers’ surplus, when agents behave like monopolists, trying to exert perfect price discrimination

7 If we restrict to economies in which CE consumptions are strictly interior, the market maker can be dispensed with.

2 The First Version: Optimistic Conjectures and Equilibrium Points

Let N = {1, , n} be the set of agents who trade in k commodities Each agent i ∈ N has an initial endowment e i ∈ IR k

+ \ {0} and a preference

relation > ∼ i on IRk+ that is convex, continuous and monotonic (in the sense

that x ≥ y, x 6= y implies x  i y) We assume that P

i∈N

e i À 0, i.e every

named commodity is present in the aggregate

An agent may enter a market either as a buyer or a seller, and submit

to each of the k commodity markets a marginal demand or supply curve.

Formally, let

M+ = {f : IR+ → IR++| f is continuous and non-decreasing}

M − = {f : IR+ → IR++| f is continuous and non-increasing}.

Then a strategic choice σ i of agent i is given by

j The supply curve has an analogous meaning Denote

σ ≡ (σ1, , σ n ) and let S σ

j , D σ

j be the aggregate supply, demand curves

We suppose that agent i acts under the optimistic conjecture that he

can exert perfect price discrimination, i.e., that he can sell (buy) starting atthe highest (lowest) prices quoted by the buyers (sellers) This means that

agent i calculates his receipts (or expenditures) on the market j as the gral, starting from 0, under the curve D σ

inte-j (or S σ

j) The generally non-convex

budget-set B i (σ) for σ = (σ1, , σ n), is then obtained by the requiring that(perceived) expenditures do not exceed (perceived) receipts, i.e.,

j < 0) means that i buys (sells) j )

The collection of strategic choices σ will be called an equilibrium point (EP) if there exist trade vectors t1, , t n in IRk such that

(i) e i + t i is > ∼ i -optimal on B i (σ) for i = 1, , n

at the same time, in equilibrium all trades compatible with the submittedstrategies are actually carried out

An EP will be called active if there is positive trade in each market.

First let us establish that at an active EP all trade T j := P

i:t i

j >0

t i

any commodity j takes place at one price, p j

Lemma 1 The curves S σ

j and D σ

j coincide and are constant on [0, T j ] at

any EP

Proof For any j, let G j := {i : t i

Z

0

S σ j

(2) and (3) together imply:

In view of the Lemma 1 we can talk not only of the allocation but also the

prices produced at an active EP These are the constant values of S σ

W.l.o.g fix i = 1, put

δ j := min[|t1

j |, T l : j ∈ J1∪ J2, l ∈ J3]

N j := {α ∈ IR : |t1j − α| < δ j }

F j := E j − R j

(Since the EP is active, δ j > 0) Now we claim, for j = 1, , k:

F j is continuously differentiable and strictly increasing on N j

1) = p1 > 0, it follows from the implicit

func-tion theorem that there is a neighborhood V of (t1

2, , t1

k ) in N2× × N k

such that if (t2, , t k ) ∈ V then there is a unique t1 which satisfies the

equation F1(t1) + + F k (t k) = 0 Thus we have an implicit function

G(t2, , t k ) = F1−1 (−F2(t2) − − F k (t k)) defined on V which is clearly

continuously differentiable Finally the point t1 = (t1

1, , t1

k) belongs by

construction to the smooth hypersurface M = {(G(t2, , t k ), t2, , t k) :

(t2, , t k ) ∈ V } ⊂ B1(σ) and, by (8), the tangent plane H to M at this point has normal p

Since we are at an EP, e1+ t1 is> ∼1-optimal on (e1+ M) ∩ IR k+ Suppose

that there is some x ∈ H+ := (e1 + t1 + H) ∩ IR k+ such that x Â1 e1+ t1

By continuity of  ∼1 we can find a neighborhood Z of x (in IR k+) with the

property: y ∈ Z ⇒ y Â1 e1+ t1 But since M is a smooth surface there exists

a point y ∗ in Z, such that the line segment between y ∗ and e1 + t1 pierces

e1+ M at some point z ∗ ∈ (e1+ M) ∩ IR k+ (see Fig.1) By convexity of> ∼1, we

have z ∗ Â1 e1+t1, contradicting that e1+t1 is> ∼1-optimal on (e1+M)∩IR k

+

We conclude that e1+t1 is> ∼1-optimal on H+ But we have e1 ∈ H+(simply

set trades to be zero, i.e., pick −t1 in H) Therefore, in fact, H+ = B1(p) Since the choice of i = 1 was arbitrary, the proposition follows.

Insert Figure 1 approximately here!

Proposition 2 If the trades t1, , t n and prices p À 0 are Walrasian, then they can be achieved at an EP

Proof For any i let

Then it is readily checked that these strategies constitute a EP and produce

the trades t1, , t n at prices p

Wal-ras Equilibria with an Infinitesimal Market Maker

The foregoing analysis can be recast in terms of strategic (Nash) equilibria(SE) of a market game Of course it is well known8 (see Maskin (1999))that CE cannot be implemented as SE unless CE consumptions are strictly

i ∈ N), one can consider a smaller domain of economies on which interiority

is guaranteed But we shall place no such restrictions here Instead we

shall imagine a “market maker”who has inventory of ε j > 0 units of each

commodity j ∈ K ≡ {1, , k} and who is ready to bring them to market

if any seller reneges on his promise to deliver, thereby giving the buyers

something to look forward to No matter how small ε = (ε1, , ε k) is, solong as it is positive, CE are implemented as SE The market maker is notcalled upon to take any action at the SE of our strategic game He only

lurks in the background It is enough for every agent i to believe that the market maker would make available the infinitesimal inventory ε, were i to

unilaterally deviate from SE and thereby trigger a situation in which some

sellers of commodity j are unable to deliver on their promises The belief in

the market maker ensures that he is never called upon to prove his existence9

(somewhat akin to the Federal Reserve’s guarantee of private banks, whichdeters bank runs and eliminates the need for the Federal Reserve to make

8 We are grateful to Stephen Morris and Andrew Postlewaite for references to the anism design literature.

mech-9 Indeed, we can reinterpret the scenario in terms of “refined” SE of the game without

the market maker, relegating the market maker to ε-trembles in the refinement (See

Section 3.7).